There are numerous influencing factors of the risk consequences
of dam break. The scientific and reasonable index system and its weight
distribution are some of the key elements for comprehensive evaluation of the
dam break risk. Taking into consideration 20 factors, including hazards,
exposure and vulnerability, the evaluation index system of the
consequences of dam break risk is constructed. Using the Statistical
Cloud Model (SCM)
to improve the entropy method, we establish the weight
calculation model of the influencing factors of dam break risk
consequences. The results shows that the top five factors with the highest
weight are risk population, flood intensity, alert time, risk understanding
and distance from the dam. Compared to traditional algebraic weight
calculation methods, the result is basically consistent with the algebraic
weight distribution, and increases the range by 2.03 times, supporting a more
scientific basis for recognizing and evaluating dam break risk
consequences.

Comprehensive evaluation of the risk consequences of dam break is the
overall description of the severity of the consequences of dam failure
(Ling et al., 2009). The factors that can influence the risk consequences of
dam break are usually composed of three factors, namely, hazards, exposure and
vulnerability (Zhang et al., 2006; Smith, 2013). The vulnerability factor can be
further divided into four aspects: loss of life, economic loss, social impact
and environmental impact. From the point of view of system science, the dam
break flood disaster system is a dynamic system with high dimensionality,
complexity and uncertainty (Ge et al., 2017). It correlates with the
development trend in risk assessment research “from low dimensional
linearity to complex high-dimensional nonlinearity”, “from single scale to
multi-dimensional space–time scale”, “from single scenario to combined
scenario” and “from certainty to uncertainty” (Zou et al., 2013).

Previous research on the index system of risk consequences and its
weight is not sufficient. The uncertainty of the impact of dam failure is
explored and suggestions for the research index system are given (Lee and Noh,
2003; Wagenaar et al., 2016). The relationship among hazard-influencing factors and relationship between exposure and vulnerability
factors are very complicated and the different types of flood including dam
break flood can cause different degrees of life loss (Jonkman et al., 2018;
Wisner and Uitto, 2009). The indirect loss index for natural disasters is
introduced and their weight is calculated using the traditional algebraic
method (Daniell et al., 2018). The DAMBREAK computer program is utilized
to analyze the downstream environmental impact and present 21 influence
receptors, but the weight distribution of them is too average (Colomer Mendoza and
Gallardo, 2008). The Statistical Cloud Model (SCM) is used for
qualitative and quantitative transformation to analyze regional water
safety systems, but it is not combined with the weight calculation (Ren et al.,
2017). In the quantitative evaluation of risk consequences, we need to
consider the combined effects of various factors, in which weight is key. The function of weight is to coordinate and balance the
difference among the indexes. It is a measure to unify each index without
considering the dimension difference among the indexes. In order to
evaluate the risk consequences more comprehensively and objectively, many
influencing factors are needed. However, too many indicators, more than nine
for example, will bring such problems as difficulty in expert scoring
and consistency testing and too average of a weight distribution.

In the course of calculating the weights, different methods have their own
emphasis. For example, the entropy weight method as one of the important methods
of weight calculation, does not adequately consider the subjective opinions
of experts. The analytic hierarchy process (AHP) is faced with the
difficulty of consistency checking when dealing with the conditions of
multiple factors (more than nine) (Su et al., 2016). When previous studies
used the data of SCM to calculate weights, they
had neglected the entropy when applying the SCM to convert subjective
opinions, resulting in the imperfection of information utilization (Mithas
et al., 2011; Wan et al., 2015). These mentioned defects all lead to lack of
scientificity in the calculation of weight. This paper introduces
SCM, which can reflect the fuzziness and randomness, to improve the entropy
method for analyzing the weight of influencing factors of dam break risk
consequences. The scientific influencing factors' weight will provide an
important basis for further research on the dam break risk comprehensive
evaluation and for the establishment and improvement of dam risk management
theory.

2.1 Risk index system

The establishment of an evaluation index system is a systematic process.
A scientific and reasonable evaluation index system is the guarantee for
accurate risk assessment of dam failure, and the evaluation result is
helpful for later research. Influencing factors of dam break risk
consequences are many and complicated in both quality and quantity, direct
and indirect contribution, and natural and social effects (Zhou et al., 2014). We
choose representative indicators as much as possible to reduce the
mutual influence and derivative of the indicators. For example, the risk
population is the most direct factor of life loss; we only set it in
life-bearing bodies, even though it influences the economic and social
aspects, but an indirect and less crucial way (Dutta et al., 2003). In the
selection of economic impact factors, the selection of GDP (gross
domestic product) per capita can better reflect the economic situation of
the dam area. Compared with the GDP of the area, it is more accurate.
Similarly, some crucial comprehensive indicators have also been selected,
for example, flood intensity parameters that directly destroy
downstream of the dam, and water environment and soil environment, which respectively
refer to the quality of water and soil after being washed by a dam break
flood. Another important indicator is the comprehensive ability and
social carrying capacity, which include the performance of downstream
disaster response, disaster rescue and relief capacity, and post-disaster
reconstruction capacity. Whether the established index system is scientific
and reasonable is directly related to whether it can objectively reflect the
nature of the vulnerability itself. On the basis of aforementioned factors
and characteristics of the dam break flood system, we establish the risk
influencing factor index system scientifically and reasonably as shown in
Fig. 1.

2.2 Weight-calculating model based on SCM-improved entropy method

Uncertainty is an intrinsic property of the objective world. The most
important and most common uncertainties include fuzziness and randomness
(Ragas et al., 2010). The influencing factor system of the dam failure risk
consequence is a multilevel and multi-index system with uncertainties (Li
et al., 1995). In determining the importance of each risk factor to the
comprehensive evaluation of the consequence, a “quantitative
conversion” of the uncertainty of the indicator is needed. In the process of
conversion, the expert's judgment makes a choice among many different
factors that mutually affect each other and will absolutely lead to the
ambiguity of boundaries, which is the fuzziness. Conversely, the risk
factors of dam break involve many aspects of life, economic loss, and
environmental and social impacts. In order to avoid the impact of experts'
personal experience and subjective factors in evaluation results, the
risk factors of dam break need to adopt the method of group decision-making. When an expert judges diverse risk factors, other experts will have different opinions, reflected in the randomness of judgments.
Therefore, the dam break risk assessment system is a complex system integrating
fuzziness and randomness. The SCM is invented under this context of
random and fuzzy features of the dam break risk system. It describes the
concept of clouds, reflects the randomness and fuzziness of
concepts in natural language, and realizes the conversion between
qualitative and quantitative information (D. Wang et al., 2016; Liu et al.,
2018). In the process of group decision-making, the traditional method is
only a simple algebraic operation of an expert's ratings, which could not
reflect the disagreements of different experts and the concentration of
opinions. In fact, experts' opinions are actually a rounded value that
focuses on a certain degree of swing, which uses a stable tendency of
the random number instead of the exact value, basically consistent with the
central idea of SCM and the concept of entropy (Yari and Chaji, 2012; M. Wang
et al., 2016).

2.2.1 SCM theory

The SCM, which was proposed by Li Deyi, is a model of uncertainty
transformation between a qualitative concept and quantitative numerical
representation (Li et al., 1995, 2004). It mainly reflects the fuzziness and
randomness of the concept of things or human knowledge in the objective world
and integrates these two together. Constituting the mutual mapping between
qualitative and quantitative, the cloud generator is the key to
the SCM's practical application.

Membership cloud. Suppose a universe U={x}; L is
the language value of the link in U. The membership degree RL (x) of the
element x in U to the qualitative concept expressed by L is a stable random
number. The membership degree distributed in the universe of discourse is
called the membership cloud as shown in Fig. 2.

The x and y axes are for the expectation number and probability of
distribution, respectively. RL (x) takes a value between 0 and 1, whereas
the cloud represents the mapping from the universe U to the interval [0,1],
that is, RL (x): U→[0,1], ∀x∈U, x→RL
(x).

It can be seen that the qualitative concept to the quantitative value on the
universe U is a one-to-many mapping relation, rather than a one-to-one
relationship on the traditional fuzzy function. The degree of membership of
x to L is a probability distribution, not a fixed value. SCM uses the
expectation (Ex), entropy (En) and hyper-entropy (He) as a whole to characterize
an uncertain concept.

Expectation (Ex). The mathematical expectation of cloud drop distribution in
the universe of discourse, that is, the domain value corresponding to the
centric of the area under the coverage of the membership cloud, is the
domain value x of the degree of membership. Generally, it is the point most
capable of characterizing the qualitative concept, reflecting the
information center value of the corresponding fuzzy concept.

Entropy (En). En is a measure of the ambiguity of a qualitative concept,
reflecting the range of values that can be accepted by the concept in the
universe U. In the SCM, entropy is mainly used to measure the ambiguity and
probability of qualitative concepts, reflecting the uncertainty of
qualitative concepts. The larger the En is, the larger the range of values that can
be accepted by the concept and the more obscure the concept is. It embodies
the flexibility of qualitative language.

Hyper-entropy (He). The measure of En uncertainty, entropy of entropy, reflects
the discreteness of cloud drops. When the He is larger, the dispersion of
cloud droplets is greater, that is, the greater the randomness of the
membership value is and the greater the “thickness” of the cloud can be.
When it is closer to the concept center or away from the center, the
randomness is relatively small, which is similar to a person's subjective
feelings.

Cloud generator. The generator is the most basic cloud algorithm, which can
achieve quantitative range and distribution rules from the qualitative
information expressed in language value. Cloud generators are mainly divided
into the forward cloud generator and the backward cloud generator. The
conversion process from qualitative concept to quantitative representation
is conducted in the forward cloud generator; the conversion process from
quantitative representation to qualitative concept is produced by the
backward cloud generator.

2.2.2 Entropy method

The subjective weight analysis method is more dependent on the experts'
opinions, and the consistency test under many factors is very difficult
(Yari and Chaji, 2012). Therefore, this paper introduces the entropy weight
method as an objective weight calculation method. Entropy is a measure of
uncertainty or randomness in information theory (Ouyang and Shi, 2013). In
general, the more uncertain or random the event is, the more information it
will contain, so the bigger the entropy is. Therefore, the most important part
of the entropy method is obtaining the differences in information, which is
the degree of variation (Wang and Chen, 2016). According to the degree of
variation of each index, we can calculate the entropy of each factor and
then use the entropy to adjust the weight of it, and finally the objective
weight value of the factors in the system is obtained.

The contribution of the numerical value in high frequencies or common
consensus factor to the qualitative concept is greater than that of the
numerical value in low frequencies (Yang and Nataliani, 2018). The En in the
SCM could coincide with the idea of the entropy method in essence (T. Wang, 2015; Dong et al., 2010). This paper makes use of similar
connotations of the SCM and entropy methods. The objective advantages of the
entropy method need to be based on large numbers of score samples, which can
be produced by the SCM cloud generator to obtain enough samples from limited
expert opinions. This paper attempts to use the SCM of the
qualitative–quantitative conversion model to improve the entropy method and
make a scientific and objective response to the weight of risk factors.

2.2.3 Improved entropy method based on SCM

Suppose there are n indicators (column vectors) and m experts (row
vectors). Each indicator computes the expectation and variance according to
the cloud model. The statistical equation for calculating the jth
indicator is as follows (Li et al., 1995).

The weight equation for the indicator calculated with the use of the
conventional algebraic method is as follows:

(4)ωj=Exj∑j=1nExjj=1,…,n.

This algebraic method is easy to use, but it does not make any use of the
changes of En in the SCM and may be misleading. For example, when the average
scores of all indicators are the same, the weight of each indicator will
calculate the same result. However, Enj and Hej could change greatly
but will not make enough reflection of the change in the original equation, so
an improved model is needed to replace this equation, as follows:

(5)ωj^=Exjln⁡1+Enj+1⋅1∑j=1nExjln⁡1+Enj+1(Enj≠0)Exj∑j=1nExj(Enj=0).

If the Enj is not equal to 0, the equation of the weight is revised and
the cloud entropy is involved in the calculation. The larger the cloud
entropy, the more divergence of opinions the expert has on the index, so the
weight of the index should be reduced. The smaller the entropy is, the
smaller the expert's disagreement on the indicator, so the weight of the
indicator should be increased. When the minimum entropy Enj is equal to
0, indicating that the indicators of the experts have the same score,
then the weight of the equation remains unchanged.

3.1 Expert scoring

According to the requirement of data volume based on the entropy method, we
invited 20 experts to score the index system. Each index scoring adopts a 100
integral point system, according to the importance without any comparison
among each other. Scoring points should be scored from the perspective of
comprehensive assessment of the risk consequences of the dam break in the
same magnitude. This scoring method can obtain the most intuitive opinion from the
expert without implying any preference of the factors and makes the scoring
process easier. In accordance with the result of the score obtained by the
backward cloud generator (Xu, 2016) based on Eqs. (1) to (3),
Exj, Enj and Hej are obtained. In order to reflect the model
characteristics of expert scoring more intuitively, the outstanding
advantage of SCM, we present the sketch map of these 20 factors' membership
cloud as in Fig. 3.

As shown in Fig. 3, the center vertex of the cloud is Ex, En represents the
width of the cloud and He represents the degree of dispersion of cloud
distribution, that is, the thickness of cloud lines. For instance, the
closer Ex is to the right side of the axis, the higher the experts' score.
The En of Lv3 is larger than that of Lv1; we can find the cloud is wider, and
the He of E4 is larger than that of E2, so the distribution of the cloud is
obviously thicker than E2. Thus, the membership cloud can obviously
reflect the degree of divergence and randomness of expert opinions.

3.2 Weight calculation

After the result of the scoring is processed by the backward cloud generator
according to Eqs. (1) to (3) and (5), the improved weight distribution
result and result comparing the algebraic method are shown in Table 1.

3.3 Discussion

In order to verify the validity of the method, the results of the
distribution contrast of the original and improved methods are drawn as Fig. 4.

According to Figs. 3, 4 and Table 1, the analysis of figures shows the
following.

The top rankings have not changed after the adjustment and still maintain
the consistency of ranking. All of these top-ranking factors are scored
higher and the opinions are concentrated, which is in line with the
objective situation. At the same time, the range increased by 2.04 times,
avoiding the problem of decentralization of weight distribution.

The distribution of weights basically corresponds to the numerical
value of Ex, reflecting the opinions of experts. At the same time, according
to the adjustment of En, which reflects the difference of expert opinion, the
weight of opinion-unified index is further enlarged. Several factors
reduced the weight due to the large differences in opinions and the further
reduction in adjusted weights. This reflects the validity of the entropy
method in handling the weight distribution through the differences in
opinions.

Thus, it can be seen that the SCM-improved entropy weight model is more in
keeping with the general cognition of the people while ensuring the
objective and fair data.

Dam break is a kind of low-probability and high-loss risk event with
uncertainties. In this paper, risk factors are divided into hazards,
exposure and vulnerability factors, and 20 factors are selected as the
main influencing factors of dam break risk consequences. We used SCM to
improve the entropy method, based on the idea that these two methods
are dealing with the divergence. The fuzziness index of the information is
generated by the backward cloud generator and then applied to the improved
formula of the entropy weight calculation model. We establish the weight
calculation model of influencing factors of dam break risk. The results
indicate that (1) the result of weight calculation conforms to expert
cognition; the main factors' weight ranking is basically consistent with the one
calculated with the traditional algebraic method. (2) Under the condition of
20 factors, the average problem of weight distribution is overcome; the
difference between the maximum and the minimum is 2 times larger. (3) This
model has the advantages of extensive applicability, benefiting from the
flexibility of index selection and the independence of expert scoring. The
method can be applied not only to the weighting analysis of risk factors
before the dam break, but also to the analysis of disaster loss after the
dam break through the targeted selection of indicators. Meanwhile, in view
of the commonality of risk indicators, experts from different countries can
obtain the weight distribution applicable to them according to their
specific tendencies. The understanding of weight can help stakeholders to
take more targeted measures to control risk factors and to allocate the
reinforcement fund more reasonably, thereby improving the effect of risk
control and risk management. In a word, it is reasonable and feasible to
apply this improved model to the weight analysis of dam break risk factors,
providing a solid foundation for risk assessment and risk management theory.

This work was funded by National Natural Science Foundation of China (grant
nos. 51679222, 51709239, 51379192), China Postdoctoral Science Foundation
(grant no. 2018M632809), Science and Technology Project of Henan Province of
China (grant no. 182102311070), Key Project of Science and Technology
Research of Education Department of Henan Province of China (grant no.
18A570007) and Science and Technology Project of Water Conservancy of Henan
Province of China (grant no. GG201813). We also thank the four reviewers and
referees for insightful comments that improved an earlier version of this
paper.

Wisner, B. and Uitto, J.: Life on the Edge: Urban social vulnerability and
decentralized, Citizen-Based Disaster Risk Reduction in Four Large Cities of
the Pacific Rim, Springer, Berlin, Heidelberg, 2009.

It is necessary to analyze the weight of multiple factors in the risk consequence of dam break. When the number of influencing factors exceeds 10, the analysis of its weight will become very difficult. In this paper, the cloud model, an artificial intelligence calculation method, is used to transform the subjective factors into a large number of data for the improved entropy weight method. The result is objective and reasonable, providing a new way of analyzing multi-factor weights.

It is necessary to analyze the weight of multiple factors in the risk consequence of dam break....